Scalar triple product
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Scalar Triple Product
The scalar triple product is any expression of the form [Equation goes here - download the original to see it.] To understand what this expression "means" we need to revise firstly the meaning of the cross product. The vector [Equation goes here - download the original to see it.] is perpendicular to both the vectors b and c. If q is the angle between these two vectors, b and c, then [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] he expression [Equation goes here - download the original to see it.] denotes the "height" of the vector c. [Diagram goes here - download the original to see it.] The area of a parallelogram is [Equation goes here - download the original to see it.] Here the base is b and the height is , so [Equation goes here - download the original to see it.] is the area of the parallelogram with sides of length [Equation goes here - download the original to see it.]. [Diagram goes here - download the original to see it.] Now imagine that we have three vectors, a, b and c. [Diagram goes here - download the original to see it.] The magnitude of the vector [Equation goes here - download the original to see it.] , that is [Equation goes here - download the original to see it.] , is the area of the base of this figure, the parallelogram with b and c as sides. [Diagram goes here - download the original to see it.] The vector [Equation goes here - download the original to see it.] is perpendicular to both b and c, and let a be the angle between this vector and the vector a. [Diagram goes here - download the original to see it.] So, from the definition of the scalar (dot) product [Equation goes here - download the original to see it.] so [Equation goes here - download the original to see it.] The quantity is the projection of the vector a onto the vector [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] So the quantity [Equation goes here - download the original to see it.] is the volume of the quasi-rectangular object bounded by vectors a, b and c and other vectors parallel to these [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] is the area of the base, and [Equation goes here - download the original to see it.]is the height. This figure is called a parallelipiped. It is a polyhedron with six faces, each of which is a parallelogram.
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Contents of
Scalar triple product
1 Scalar Triple Product
2 The triple scalar product as a "signed" volume
3 Shortest distance between two skew lines
4 Intersecting and parallel lines
5 Co-planar points
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